The embedded Nash problem for a hypersurface in a smooth algebraic variety is to characterize geometrically the maximal irreducible families of arcs with fixed order of contact along the hypersurface. We show that divisors on minimal models of the pair contribute with such families. We solve the problem for unibranch plane curve germs, in terms of the resolution graph. These are embedded analogs of known results for the classical Nash problem on singular varieties.

We develop a Thom–Mather theory of frontals analogous to Ishikawa's theory of deformations of Legendrian singularities but at the frontal level, avoiding the use of the contact setting. In particular, we define concepts like frontal stability, versality of frontal unfoldings or frontal codimension. We prove several characterizations of stability, including a frontal Mather–Gaffney criterion, and of versality. We then define the method of reduction with which we show how to construct frontal versal unfoldings of plane curves and show how to construct stable unfoldings of corank 1 frontals with isolated instability which are not necessarily versal. We prove a frontal version of Mond's conjecture in dimension 1. Finally, we classify stable frontal multigerms and give a complete classification of corank 1 stable frontals from $\mathbb {C}^3$ to $\mathbb {C}^4$.

Let f be an isolated singularity at the origin of $\mathbb {C}^n$. One of many invariants that can be associated with f is its Łojasiewicz exponent $\mathcal {L}_0 (f)$, which measures, to some extent, the topology of f. We give, for generic surface singularities f, an effective formula for $\mathcal {L}_0 (f)$ in terms of the Newton polyhedron of f. This is a realization of one of Arnold’s postulates.

We study the Hodge and weight filtrations on the localization along a hypersurface, using methods from birational geometry and the V-filtration induced by a local defining equation. These filtrations give rise to ideal sheaves called weighted Hodge ideals, which include the adjoint ideal and a multiplier ideal. We analyze their local and global properties, from which we deduce applications related to singularities of hypersurfaces of smooth varieties.

We undertake a systematic study of Lipschitz normally embedded normal complex surface germs. We prove, in particular, that the topological type of such a germ determines the combinatorics of its minimal resolution which factors through the blowup of its maximal ideal and through its Nash transform, as well as the polar curve and the discriminant curve of a generic plane projection, thus generalizing results of Spivakovsky and Bondil that were known for minimal surface singularities. In an appendix, we give a new example of a Lipschitz normally embedded surface singularity.

We consider the relative Bruce–Roberts number $\mu _{\textrm {BR}}^{-}(f,\,X)$ of a function on an isolated hypersurface singularity $(X,\,0)$. We show that $\mu _{\textrm {BR}}^{-}(f,\,X)$ is equal to the sum of the Milnor number of the fibre $\mu (f^{-1}(0)\cap X,\,0)$ plus the difference $\mu (X,\,0)-\tau (X,\,0)$ between the Milnor and the Tjurina numbers of $(X,\,0)$. As an application, we show that the usual Bruce–Roberts number $\mu _{\textrm {BR}}(f,\,X)$ is equal to $\mu (f)+\mu _{\textrm {BR}}^{-}(f,\,X)$. We also deduce that the relative logarithmic characteristic variety $LC(X)^{-}$, obtained from the logarithmic characteristic variety $LC(X)$ by eliminating the component corresponding to the complement of $X$ in the ambient space, is Cohen–Macaulay.

Vanishing cycles, introduced over half a century ago, are a fundamental tool for studying the topology of complex hypersurface singularity germs, as well as the change in topology of a degenerating family of projective manifolds. More recently, vanishing cycles have found deep applications in enumerative geometry, representation theory, applied algebraic geometry, birational geometry, etc. In this survey, we introduce vanishing cycles from a topological perspective and discuss some of their applications.

We prove the Lipman–Zariski conjecture for complex surface singularities with $p_{g}-g-b\leqslant 2$. Here $p_{g}$ is the geometric genus, $g$ is the sum of the genera of exceptional curves and $b$ is the first Betti number of the dual graph. This improves on a previous result of the second author. As an application, we show that a compact complex surface with a locally free tangent sheaf is smooth as soon as it admits two generically linearly independent twisted vector fields and its canonical sheaf has at most two global sections.

We compute the Hodge ideals of $\mathbb{Q}$-divisors in terms of the $V$-filtration induced by a local defining equation, inspired by a result of Saito in the reduced case. We deduce basic properties of Hodge ideals in this generality, and relate them to Bernstein–Sato polynomials. As a consequence of our study we establish general properties of the minimal exponent, a refined version of the log canonical threshold, and bound it in terms of discrepancies on log resolutions, addressing a question of Lichtin and Kollár.

For a line arrangement ${\mathcal{A}}$ in the complex projective plane $\mathbb{P}^{2}$, we investigate the compactification $\overline{F}$ in $\mathbb{P}^{3}$ of the affine Milnor fiber $F$ and its minimal resolution $\tilde{F}$. We compute the Chern numbers of $\tilde{F}$ in terms of the combinatorics of the line arrangement ${\mathcal{A}}$. As applications of the computation of the Chern numbers, we show that the minimal resolution is never a quotient of a ball; in addition, we also prove that $\tilde{F}$ is of general type when the arrangement has only nodes or triple points as singularities. Finally, we compute all the Hodge numbers of some $\tilde{F}$ by using some knowledge about the Milnor fiber monodromy of the arrangement.

We extend the infinitesimal Torelli theorem for smooth hypersurfaces to nodal hypersurfaces.

Given an $n$-dimensional variety $Z$ with rational singularities, we conjecture that if $f:Y\rightarrow Z$ is a resolution of singularities whose reduced exceptional divisor $E$ has simple normal crossings, then

$$\begin{eqnarray}\displaystyle R^{n-1}f_{\ast }\unicode[STIX]{x1D6FA}_{Y}(\log E)=0. & & \displaystyle \nonumber\end{eqnarray}$$

$Z$

$D$

$n$

$X$

$\mathscr{O}_{X}(\ast D)$

$n-3$

We define and study twisted Alexander-type invariants of complex hypersurface complements. We investigate torsion properties for the twisted Alexander modules and extend the local-to-global divisibility results of Maxim and of Dimca and Libgober to the twisted setting. In the process, we also study the splitting fields containing the roots of the corresponding twisted Alexander polynomials.

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of such a hypersurface are heavily constrained. We introduce the notion of an elliptic point of a log symplectic structure, which is a singular point at which a natural transversality condition involving the modular vector field is satisfied, and we prove a local normal form for such points that involves the simple elliptic surface singularities $\widetilde{E}_{6},\widetilde{E}_{7}$ and $\widetilde{E}_{8}$. Our main application is to the classification of Poisson brackets on Fano fourfolds. For example, we show that Feigin and Odesskii’s Poisson structures of type $q_{5,1}$ are the only log symplectic structures on projective four-space whose singular points are all elliptic.

Using a recent result by S. Papadima and A. Suciu, we show that the equivariant Poincaré– Deligne polynomial of the Milnor fiber of a projective line arrangement having only double and triple points is combinatorially determined.

We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

We study symplectic deformation types of minimal symplectic fillings of links of quotient surface singularities. In particular, there are only finitely many symplectic deformation types for each quotient surface singularity.

The order of the Milnor fiber monodromy operator of a central hyperplane arrangement is shown to be combinatorially determined. In particular, a necessary and sufficient condition for the triviality of this monodromy operator is given.

It is known that the complement of a complex hyperplane arrangement is cohomologically Tate and, if the arrangement is defined over ℚ, has polynomial count. We show that these properties hold for the corresponding Milnor fibers if the monodromy is trivial.

We construct a hyperplane arrangement defined over ℚ, whose Milnor fiber has a nontrivial monodromy operator, is cohomologically Tate, and has no polynomial count. Such examples are shown not to exist in low dimensions.

We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold’s strange duality between the 14 exceptional unimodal singularities.